LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION - STATISTICS
THIRD SEMESTER – November 2008
BA 24
ST 3808 - MULTIVARIATE ANALYSIS
Date : 03-11-08
Time : 9:00 - 12:00
Dept. No.
Max. : 100 Marks
PART – A
Answer all the questions.
(10 X 2 = 20)
1. If X and Y are two independent standard normal variables then obtain the distribution of two times
of the mean of these two variables.
 0  1 0.8
2 If X = ( X1, X2 )'N2   , 
  then obtain the conditional distribution of X1 given
 0  0.8 1
X2=x2.
3. Suggest unbiased estimators for  and  when the sample is from N( ,  ).
4 . What is meant by residual plot?.
5. Explain the use of partial and multiple correlation coefficients.
     xx  xy 
6. Let(xi ,yi )', i=1,2 be independently distributed each according to N2    , 
 
    xy  yy 
Find the distribution of ( X Y ) )'.
7. Define Fisher's Z-transformation.
8. In a multivariate normal distribution, show that every linear combination of the
component of the random vector is normal. Is the converse true?.
9. Explain how canonical correlation is used in multivariate data analysis.
10. Explain classification problem into two classes.
PART B
Answer any FIVE questions.
(5 X 8 = 40)
11. Find the multiple correlation coefficient between X1 and X 2 , X3, … , X p.
Prove that the conditional variance of X1 given the rest of the variables cannot be
greater than unconditional variance of X1.
12. Derive the characteristic function of multivariate normal distribution
13. Let the correlation matrix be given by
1   
R=   1  
,0.
   1
Obtain the principal components.
14. Let x be an observation from one of the populations N(1 , 1 ) and N(2 , 2 ). Explain
how will you classify x into one of the populations.
15. Giving a suitable example describe how objects are grouped by complete linkage
method.
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16. Let X  Np (, ). If X(1) and X(2) are two subvectors of X, obtain the conditional
distribution of X(1) given X(2).
17. Prove that the extraction of principal components from a dispersion matrix is the
study of characteristic roots and vectors of the same matrix .
18. Derive the procedure to test the equality of mean vectors of two p-variate
normal populations when the dispersion matrices are equal.
PART C
Answer any two questions.
(2 X 20 = 40)
19. a) Define generalized variance
b) Show that the sample generalized variance is zero if and only if the rows of the matrix
of deviations are linearly dependent.
c) Find the covariance matrix of the multinormal distribution which has the quadratic
form
2x12+x22+4x32-x1x2-2x1x3.
( 3+12+5)
20.a) Define i) Common factor ii) Communality iii) Total variation
b) Explain the principal component ( principal factor )method of estimating L
in the factor analysis method.
c) Discuss the effect of an orthogonal transformation in factor analysis method.
( 6+8+6)
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21. a) Explain in detail T -Statistic with an illustration.
b) Distinguish between classification and discrimination with an illustration.
(10+10)
22. What are canonical correlations and canonical variables? .Describe the extraction
of canonical correlations and their variables from the dispersion matrix. Also
show that there will be p canonical correlations if the dispersion matrix is of size p.
( 2+10+8 )
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